A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings

By Tomonari Suzuki. 2006

121 citations recorded by the source corpus.

In this paper, we prove a Halpern-type strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5d03559230ff8cf301aa6e4ba822fc161216421b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

In Numerical Analysis

Citations in the corpus

Effective metastability of Halpern iterates in CAT(0) spaces (2011, background): “Recently, Chidume and Chidume [10] and Suzuki [46] proved that if the nonexpansive mapping T in (9) is averaged, then (C1) and (C3) suffice for obtaining the convergence of (xn).”
Solving the split feasibility problem without prior knowledge of matrix norms (2012, background): “It is well known that Halpern’s algorithm [13] has a strong convergence for finding a fixed point of a nonexpansive mapping; see [15, 26, 29–31] and the recent survey [16].”
Weak and Strong Convergence of Algorithms for the Split Common Null Point Problem (2011, background): “Remark 3.3 Observe that in Theorem 3.2 we assume that γ ∈ (0, 2/L), while in [20, Theorem 6.3], γ is assumed to be in (0, 1/L), which obviously was a more restrictive assumption.”
The Split Common Null Point Problem (2011, methodology): “Next we present two known theorems, the Krasnosel’skĭıMann-Opial theorem [37, 39, 44] and the Halpern-Suzuki theorem [34, 48].”